Problem: $\dfrac{ -10s - 6t }{ 4 } = \dfrac{ -2s + 10u }{ -5 }$ Solve for $s$.
Answer: Multiply both sides by the left denominator. $\dfrac{ -10s - 6t }{ {4} } = \dfrac{ -2s + 10u }{ -5 }$ ${4} \cdot \dfrac{ -10s - 6t }{ {4} } = {4} \cdot \dfrac{ -2s + 10u }{ -5 }$ $-10s - 6t = {4} \cdot \dfrac { -2s + 10u }{ -5 }$ Multiply both sides by the right denominator. $-10s - 6t = 4 \cdot \dfrac{ -2s + 10u }{ -{5} }$ $-{5} \cdot \left( -10s - 6t \right) = -{5} \cdot 4 \cdot \dfrac{ -2s + 10u }{ -{5} }$ $-{5} \cdot \left( -10s - 6t \right) = 4 \cdot \left( -2s + 10u \right)$ Distribute both sides $-{5} \cdot \left( -10s - 6t \right) = {4} \cdot \left( -2s + 10u \right)$ ${50}s + {30}t = -{8}s + {40}u$ Combine $s$ terms on the left. ${50s} + 30t = -{8s} + 40u$ ${58s} + 30t = 40u$ Move the $t$ term to the right. $58s + {30t} = 40u$ $58s = 40u - {30t}$ Isolate $s$ by dividing both sides by its coefficient. ${58}s = 40u - 30t$ $s = \dfrac{ 40u - 30t }{ {58} }$ All of these terms are divisible by $2$ $s = \dfrac{ {20}u - {15}t }{ {29} }$